 Lebesgue decomposition commutes with finite measure suprema

Take Lebesgue decompositions relative to a fixed finite positive measure $\kappa$. The finite union $S$ of null supports of their singular components is still $\kappa$-null. On $S$ each original measure equals its singular component; on $S^c$ it equals its absolutely continuous component. Restriction commutes with the partition formula defining the finite supremum of positive measures. Thus the supremum restricts to ${\bigvee }_i{\mu }_i^s$ on $S$ and to ${\bigvee }_i{\mu }_i^a$ off $S$. The latter is absolutely continuous with respect to $\kappa$ because it is bounded by the sum of the absolutely continuous components. Uniqueness of the Lebesgue decomposition proves the displayed identities, and the Radon-Nikodym derivative of the absolutely continuous component is the pointwise maximum of the individual derivatives.